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A sparse reconstruction framework forFourier-based plane wave imaging

Adrien Besson, Miaomiao Zhang, François Varray, Hervé Liebgott, Denis Friboulet, Yves Wiaux,Jean-Philippe Thiran, Rafael E. Carrillo and Olivier Bernard

Abstract—Ultrafast imaging based on plane-wave (PW) in-sonification is an active area of research due to its capabilityof reaching high frame rates. Among PW imaging methods,Fourier-based approaches have demonstrated to be competitivecompared to traditional delay and sum methods. Motivated bythe success of compressed sensing techniques in other Fourierimaging modalities, like magnetic resonance imaging, we proposea new sparse regularization framework to reconstruct highquality ultrasound images. The framework takes advantage ofboth the ability to formulate the imaging inverse problem inthe Fourier domain and the sparsity of ultrasound images ina sparsifying domain. We show, by means of simulations, invitro and in vivo data, that the proposed framework significantlyreduces image artifacts, i.e. measurement noise and side lobes,compared to classical methods, leading to an increase of the imagequality.

Index Terms—Ultrafast imaging, Fourier imaging, Sparse rep-resentation, `1-minimzation.

I. INTRODUCTION

ULTRASOUND imaging (US) has become one of the mostused imaging modalities in the last 30 years. In theconventional US systems, the number of transmitted wavesis usually equal to the number of scan lines, thus limiting theframe rates to several tens of frames per seconds. Although thisframe rate is sufficient to perform most of the applications of2D US imaging, a higher frame rate is required for the under-standing of more complex dynamics such as echocardiographyfor the heart motion analysis as well as for performing 3D or4D imaging where thousands of scan lines are necessary. Inorder to address these challenges, synthetic aperture methods[1], [2] where few transducer elements are used to sequentiallyinsonify the whole medium as well as methods based on plane-wave insonifications (PW) have thus been proposed. Methodsbased on PW transmissions use PW to insonify the wholemedium with only few transmitted waves and backscatteredechoes are processed in parallel to reconstruct many scanlines simultaneously. Using such modality, the frame rate isno longer limited by the number of scan lines but only by the

A. Besson, J.-Ph. Thiran and R. E. Carrillo are with the Signal Process-ing Laboratory (LTS5), Ecole polytechnique fédérale de Lausanne (EPFL),Lausanne, Switzerland.

A. Besson and Y. Wiaux are with the Institute of Sensors, Signals andSystems, Heriot-Watt University, Edinburgh, UK

O. Bernard, M. Zhang, F. Varray, H. Liebgott and D. Fribouletare with the University of Lyon, CREATIS, CNRS UMR5220, InsermU1206, INSA-Lyon, University of Lyon 1, Villeurbanne, France. E-mail:olivier.bernard@creatis.insa-lyon.fr

J.-Ph. Thiran is with the Department of Radiology, University HospitalCenter (CHUV) and University of Lausanne (UNIL), Lausanne, Switzerland.

time of flight of the US wave, allowing US imaging systemsto reach thousands frames per seconds opening a whole rangeof applications such as shear wave elastography [3], imagingof pulse waves [4], ultrafast Doppler imaging [5], [6], ultrafastvector flow imaging [7] and imaging of contrast agents [8].For an exhaustive presentation of the applications, one mayrefer to [9].

The development of ultrafast imaging has been intrinsicallylinked to the possibility to achieve efficient beamformingmethods. When a PW reaches an inhomogeneity in themedium (scatterer), part of its energy is backscattered andthe inhomogeneity becomes a secondary source. Assuming apropagation in a homogeneous medium with a given speedof sound, the propagation of the US wave from the secondarysource back to the surface has an hyperbolic travel-time curve.The received echo signals thus consist in a set of hyperboliccurves coming from all the inhomogeneities in the mediumand image reconstruction methods aim at inferring the positionof the scatterers from these hyperbolic profiles. In order todo so, Montaldo et al. proposed a spatial-based approach inwhich the backscattered echo signals are integrated over all thepossible hyperbolas. The value of the integral is then assignedto the corresponding point in the desired image [10]. Anotherapproach has been proposed by Lu et al. in the 90s basedon the use of limited diffraction beam theory to perform highframe rate imaging [11]–[14]. In their approach, a pulsed PWis used in transmission to reconstruct the Fourier spectrumof the desired image. They later extended their approach tovarious transmission schemes such as steered PWs (SPW) al-lowing an increase of the image quality [15]. Recently, Garciaet al. introduced an alternative to the method of Lu et al.based on a modification of the Stolt’s f-k migration techniqueyielding image quality similar to the method of Lu et al. [16].Recently, Bernard et al. proposed another alternative based onsampling the Fourier space radially and exploiting the Fourierslice theorem to retrieve the desired image spectrum, as inother imaging modalities such as computed tomography [17].

The use of ultrafast methods with only one PW leads to animage quality lower than conventional delay and sum (DAS)method with focus transmitted beams [18]. This decreasemainly comes from the fact that the transmitted energy, whilespread in the entire medium, is far lower than when it isfocused as in the conventional imaging configuration. Toaddress this problem, coherent compounding of PWs has thusbeen introduced by Montaldo et al. [10]. Based on emitting-PW with well chosen angles and adding them with differentdelay strategies, this method enables creating a synthetic focus

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in the full image range and leads to a noticeable increase of theimage quality. However, PW compounding causes a decreaseof the frame rate, proportional to the number of compoundedPWs. Moreover, at large depth, this method becomes infeasiblesince the overlap of the PW does not cover homogeneouslythe distant regions relative to the probe size.

Compressed sensing (CS) has attracted much interest inthe medical imaging community because of the potential toobtain high quality images from less data. By doing so, CSenables faster acquisition while guaranteeing similar imagequality. The ability to apply CS framework to an imagingproblem mainly relies on two pillars, namely the ability torelate the measurements to the desired image by a linearmeasurement process and the prior knowledge that imagesare sparse (or compressible) in a predefined model. Giventhese two pillars, it appears that medical imaging has char-acteristics that promote the use of CS. Firstly, the user cancontrol the acquisition scheme in order to make it suitableto CS. Secondly, most of medical imaging modalities havemeasurement models which can be described by projections.In X-ray tomographic imaging (X-ray CT), the sinogram isrelated to the attenuation coefficient by the Beer-Lambert law(see [19] for detailed information) which, when discretized,leads to a linear system, thus compliant with the CS model.Moreover, gradient magnitude image sparsity is a powerfulprior leading to an extensive number of applications of CS toX-ray CT based on total variation (TV) minimization [20]–[22]. In magnetic resonance imaging (MRI), k-space samplesof the final image are acquired and the measurement modelis an inverse discrete Fourier transform (FT). In their pioneerwork, Lustig et al. proposed to retrieve the final image frompartial Fourier data enforcing a sparsity prior both in thewavelet domain and under the TV transform [23] opening theway to fast MRI image acquisition. There has also been muchwork in applying CS reconstruction to more advanced MRtechniques such as spread spectrum MRI [24], quantitativeMR imaging [25], [26], diffusion MRI [27] and dynamic MRI[28], [29].

In the context of ultrasonic imaging, several studies havealready exploited the sparsity of backscattered echo signals inthe wave atom frame [30], as well as of radio frequency imagesin specific frames such as 2D Fourier basis [31], wavelet basis[32], or even learned dictionaries [33]. Schiffner et al. intro-duced CS-based plane wave beamforming in the frequencydomain assuming sparsity in an orthonormal wavelet basis[34] while Chernyakova et al. used a Xampling scheme anda finite rate of innovation model to achieve CS-based Fourierbeamforming [35]. In a recent paper, David et al. introducedCS-based time domain beamforming [36] in which the inversescattering problem is solved using Green’s function in anhomogeneous medium.

Motivated by the success of CS for MR imaging, weintroduce a sparse-based reconstruction of US images usingFourier-based beamforming methods. Indeed, insonifying amedium with a PW corresponds to populate the Fourier spec-trum of the desired image with non-uniform Fourier samples[14], [16], [17]. Thus, as for MRI, Fourier-based beamformingmethods aim at recovering an image from partial Fourier

measurements coming from the spectrum of the backscatteredecho signals. In order to apply a CS-based reconstruction, weformulate the imaging problem as a linear inverse problemrelating the desired image to the spectrum of the echoes by anon-uniform Fourier transform (NUFT) [37]. Then we studyseveral wavelet-based and Dirac-based models as sparsitypriors.

The paper is organized as follows. In Section II, theCS framework is briefly summarized. Section III details thethree existing Fourier-based reconstruction approaches in USimaging. In Section IV, the sparse reconstruction methodis introduced. Section V describes the different experimentsand simulations performed to evaluate the proposed methods.Experimental results of the evaluation are presented in SectionVI and a discussion is followed in Section VII. Concludingremarks are given in Section VIII.

II. COMPRESSED SENSING FRAMEWORKThe now famous theory of CS introduces a signal acqui-

sition framework that goes beyond the traditional Nyquistsampling paradigm [38]–[40]. Let x ∈ CN be the signalunder scrutiny. The fundamental premise in CS is that certainclasses of signals, such as natural images, have a conciserepresentation in terms of a sparsity dictionary Ψ, such thatx = Ψα, where most of the coefficients α are zero, or small,and only few are significant. CS demonstrates that such sparseor compressible signals can be acquired using a small numberof linear measurements and then recovered by solving a non-linear optimization problem [38]–[40].

Formally, the signal x is measured through the linear modely = Φx+n, where y ∈ CM denotes the measurement vector,Φ ∈ CM×N , M < N , is the sensing matrix and n ∈ CMrepresents the observation noise (or model inaccuracies). Re-covering x from y poses an ill-posed linear inverse problemwhere the sparse prior on the signal regularizes the solution.CS shows that the following convex problem can recover xunder certain conditions on the matrix Φ [41]:

minx̄∈CN

‖Ψ†x̄‖1 subject to ‖y − Φx̄‖2 ≤ �, (1)

where Ψ† denotes the adjoint operator of Ψ and � is anupper bound on the `2-norm of the noise. Recall that the`p-norm of a complex-valued vector a ∈ CM is defined as‖a‖p ≡ (

∑Mi=1 |ai|p)1/p, where | · | represents the modulus

of a complex number. The choice of the `1-norm instead ofthe `0-norm (real measure of sparsity) is for convex relaxationpurpose. See [40] for a thorough review on the mathematicalprinciples of CS.

III. OVERVIEW OF THE FOURIER METHODSWhen a PW is used to insonify a medium, the backscat-

tered echo signals can be beamformed in the Fourier domainusing Fourier-based beamforming methods [12], [16], [17]which share the same general scheme. The first step, calledpreprocessing, consists in applying a discrete 2D FT to thebackscattered echo signals. The intermediate image obtainedat this step is called preprocessed spectrum. The second stepconsists in relating the preprocessed spectrum to the desired

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image by means of a NUFT, i.e. a FT on a non-uniform k-space defined by each method.

In the following sections, we will briefly summarize thedifferent Fourier-based beamforming methods. As explainedabove, while the general scheme remains the same, the meth-ods differ in the non-uniform space on which the NUFT isapplied. Once this non-uniform space is defined, the desiredimage can be obtained from the preprocessed spectrum bytaking the adjoint NUFT operator [37].

Fig. 1: Notations used in the remainder of the paper.

A. Lu method

Lu et al. derive X-waves relationship to theoretically modelthe behaviour of the transducers array in transmit and receive[14].

1) Preprocessing: The preprocessing step consists in takingthe 2D FT of the backscattered echo signals. Formally, let usintroduce r (xi, t) the backscattered echo signals. Then, thepreprocessing is denoted as

y (kx, k) =

∫∫xi∈R,t∈R

r (xi, t) e−j(kxxi+kt)dxidt (2)

= F(kx,k) (r) (3)

with F(kx,k)(.) the 2D FT on the frequency node (kx, k) andk = 2π fc , with f the temporal frequency.

2) Frequency remapping: In a pulse echo configuration,if we denote by (k′x, k

′z), the k-space of the desired radio

frequency (RF) image, the following relationship holds:{kx = k

′x − k′z sin (θt) = gxL(k′x, k′z)

k =k′2x +k

′2z

2k′z cos(θt)+2k′x sin(θt)

= gzL(k′x, k′z),

(4)

where (kx, k) accounts for the k-space of the backscatteredecho signals.

Thus, the final RF image s (x, z) is related to the pre-processed spectrum by a NUFT on (kx, k) defined by theremapping (4) and the following relationship holds:

y (kx, k) = F(gxL (k′x,k′z),gzL (k′x,k′z)) {s (x, z)} . (5)

B. Garcia method

Garcia et al. propose a different approach for PW imagingbased on Stolt’s migration technique [16]. The main assump-tion is called exploding reflector model (ERM) and states thatthe scatterers in the medium all explode at the same time andbecome emitting sources. Considering the ERM wave fieldcoming from a PW with angle θt, the objective is to retrievethe value of the field at t = 0 knowing the value of the field atthe surface. Garcia et al. demonstrate that, as for Lu method,the 2D FT of the backscattered echo signals is linked to the 2DFT of the desired RF image by the remapping defined below.

1) Preprocessing: The preprocessing step, just like in Lu’smethod, consists in taking the 2D FT of the backscatteredecho signals and the preprocessed spectrum y (kx, k) is givenby (2).

2) Frequency remapping: The following relationship holdsbetween the k-space representation of the desired RF imageat (k′x, k

′z) and the preprocessed spectrum:{kx = k

′x = gxG(k

′x, k′z)

k = ĉcsign (k′z)√k′2z + k

′2x = gzG(k

′x, k′z)

(6)

with ĉ = c√1+cos(θt)+sin2(θt)

.

The RF image s (x, z) is related to the preprocessed spec-trum by applying a NUFT on the k-space defined in equation(6) and we have the following relationship:

y (kx, k) = F(gxG (k′x,k′z),gzG (k′x,k′z)) {s (x, z)} . (7)

with gxG(k′x, k′z) and gzG(k

′x, k′z) defined in (6).

C. Ultrasound Fourier slice theory

Bernard et al. demonstrate, using the Fourier slice theorem,that the temporal FT of a received PW steered with a givenangle is a radial line in the k-space of the desired image [17].Thus, by simulating several steering angles θr at receptionfor a given angle at emission, it is possible to populate thespectrum of the desired RF image to recover it.

1) Preprocessing: The preprocessing step is slightly dif-ferent than the two other methods and consists in three mainsteps for each steered angle θr [42]:

1. Apply a linear delay law Pθr on the backscattered echosignals: y1 (xi, t, θr) = Pθr {r (xi, t)}.

2. Sum the resulting signals on the lateral direction:y2 (z, θr) =

∑xi

y1 (xi, t, θr).

3. Compute the 1D temporal FT of the resulting signal:y (k, θr) = Fk (y2) with k = fc .

2) Frequency remapping: Given the preprocessed spectrumy (k, θr) defined in III-C1 and a steering angle θt at transmis-sion, Bernard et al. show that the following relationship holds:

k =k′2x + k

′2z

2 (k′x sin(θt) + k′z cos(θt))

θr = arctan(

2 k′x k′z cos(θt)+(k

′2x −k

′2z ) sin(θt)

2 k′x k′z sin(θt)+(k

′2z −k′2x ) cos(θt)

) . (8)The region of the spectrum corresponding to a fixed value

of θr is a line of angle ξr = fθt(θr) with fθt(.) =arctan((sin(θt) + sin(.))/(cos(θt) + cos(.))). Thus, for a

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range of values of θr, the spectrum of the desired im-age is populated with lines of different angles. In thesame way as Lu and Garcia methods, the final image isthen related to the preprocessed spectrum by a NUFT on(gxB (k

′x, k′z) = k sin θr, gzB (k

′x, k′z) = k cos θr) with k and θr

given by (8). We thus have:

y (kx, k) = F(gxB (k′x,k′z),gzB (k′x,k′z)) {s (x, z)} . (9)

D. Image reconstruction

Given the relationship between the desired RF images (x, z) and the preprocessed spectrum y (kx, k), the classicalway to retrieve the image from the preprocessed spectrumconsists in performing an inverse NUFT. However, y (kx, k)does not have complete information on the spectrum of s (x, z)thus recovering s from y is an ill-posed problem as it will bedescribed in the following section. In classical beamformingapproaches, the inverse NUFT is approximated by the adjointNUFT, also known as the filtered back projection method,inducing measurement inaccuracies and image artifacts.

IV. SPARSE-BASED BEAMFORMING

A. Motivation

1) Fourier-based beamforming is an ill-posed problem:When a wave is emitted from a source with limited aper-ture, Lu has demonstrated that it generates so-called limiteddiffraction array beams [13]. Plane waves have been describedas part of these limited diffraction array beams [13]. It hasalso been demonstrated that the k-space of the backscatteredecho signals corresponding to plane wave insonification islimited by evanescence properties of the waves [14]. In theparticular case of a plane wave with normal incidence, itmay be ascertained that k ≥ |kx| [15] with kx a functionof the aperture size. The k-space of the backscattered echosignals is then related to the k-space of the desired imagethrough the remapping described in Section III. Thus, due tothe evanescence properties, only part of the desired image k-space may be recovered as illustrated in Fig. 2 in [15]. Sincethe entire image spectrum cannot be retrieved, the problem isstated as ill-posed.

2) Interpolation schemes in the NUFT: Since the fastFourier transform (FFT) algorithm cannot be applied on non-uniform grids, NUFT implies an interpolation process duringthe frequency remapping, in which the non-uniform space isprojected on a uniform grid. This interpolation may createartifacts and loss of signals [37]. Usual ways to address thisproblem consists of zero-padding strategies or optimized in-terpolation schemes [37]. However, it implies a non-negligibleadditional computational cost and the quality improvementremains limited.

3) The proposed approach: The idea behind the proposedapproach is to come up with an alternative to classical ap-proaches in which the ill-posed problem is regularized byexploiting sparsity of the US images in an appropriate model.The desired image is then retrieved by solving a problemsimilar to (1).

B. Proposed sparse-based beamforming method

Since the method is based on the CS framework describedin Section II, it relies on two pillars:• The ability to pose the Fourier-based beamforming as an

inverse problem.• The sparsity of the ultrasound images in an appropriate

model.1) Problem formulation: The first pillar of the proposed

method consists in deriving a measurement operator fromthe acquisition model. Formally, if y denotes the discretizedpreprocessed spectrum defined in Section III for the differentmethods, and s denotes the desired image, the objective is toidentify Φ such that y = Φs+n, where n accounts for noiseand model perturbations.

2) Measurement operator: Notice that s is related to yby a 2D FT on a non-uniform space defined by the differentremappings described in Section III.

Let us first consider the discretization of the preprocessingstep. We define a regular grid for the backscattered echosignals, denoted as r, and the corresponding k-space grid inthe following equations:{

xi ={jp, ∀j ∈

{−Nt2 , ...,

Nt2 − 1

}}t =

{lfs, ∀l ∈ {0, ..., Nr − 1}

} (10)and kx =

{2πmp×Nt , ∀m ∈

{−Nt2 , ...,

Nt2 − 1

}}k =

{2πlfsc×Nr , ∀l ∈ {0, ..., Nr − 1}

} , (11)where p is the pitch, Nt is the number of transducer elements,fs denotes the sampling frequency, c is the speed of soundand Nr is the number of samples in the axial direction. Thediscretized preprocessed spectrum is obtained by discretizingthe different continuous operations defined in Section III for allthe methods. For Lu and Garcia methods, the preprocessingstep amounts to compute y = FRr, where FR denotes thediscrete 2D FT on the k-space defined in (11). For UFSB, thepreprocessing step to obtain y from r is described in SectionIII-C for each steering angle θr.

Let us also define a discrete uniform grid for the desired RFimage space, not necessarily the same grid as the backscatteredecho signals: x =

{mLNX

, ∀m ∈{−NX2 , ...,

NX2 − 1

}}z =

{lZmaxNZ

, ∀l ∈ {0, ..., NZ − 1}} (12)

with L the width of the probe, NX the number of imagesamples in the lateral direction, NZ the number of imagesamples in the axial direction and Zmax the maximum depth.The following grid of the corresponding image k-space can bededuced from (12):{

k′x ={

2πmL , ∀m ∈

{−NX2 , ...,

NX2 − 1

}}k′z =

{2πlZmax

, ∀l ∈ {0, ..., NZ − 1}}.

(13)

In order to take advantage of the FFT we use the non-uniform fast Fourier transform (NUFFT) to implement the 2DNUFT. The NUFFT operator can be modelled as Φ = GFS,

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where FS denotes the 2D FFT operator that computes the FTof s in the discrete k-space (k′x,k

′z) and G ∈ RNrNt×NXNZ

is a sparse matrix implementing a convolutional interpolationoperator that models the map from the discrete frequencygrid onto the continuous values (kx,k) according to thedifferent remappings described in Section III. We thus havethe following linear model:

y = GFSs+ n = Φs+ n, (14)

where n accounts for the measurement noise and modelinaccuracies induced by the interpolation.

3) The sparsifying model: The second pillar of the pro-posed method resides in the existence of a sparsifying modelΨ, i.e. a model in which the US images are compressiblemeaning that their representation in this model contains manyzeroes. In the literature, various models have already beenproposed mainly relying on wavelet-based models [32], [34],[36], [43]. However, the choice of the best model is a hardtask since it is highly dependent on the content of the image,unknown a priori. In this paper, we propose to investigatedifferent sparsifying models described below:• Dirac basis: In this very simple model, the operator Ψ

is the identity. This model is suited to images made offew sparse sources.

• Orthogonal wavelet transform: In this model, the oper-ator Ψ is the wavelet transform. This model is suited forimages with textural information.

• Undecimated wavelet transform: In this model, theoperator Ψ is a slight variation of the wavelet transformwhere each decomposition has the same size as theoriginal image [44].

• Sparsity averaging model (SA): The operator for the SAmodel is composed of the concatenation of Daubechieswavelet transforms with different wavelet mother func-tions ranging from Daubechies 1 (Db1) to Daubechies 8(Db8) as it has already been proposed in previous studies[42], [45]. Thus,

Ψ =1√q

[Ψ1, ...,Ψq] (15)

where q = 8 and Ψi denotes i-th Daubechies wavelet.Db1 is the Haar basis promoting piece-wise smoothsignals while Db2 to Db8 provide smoother sparse de-compositions.

4) The `1-minimization algorithm: The proposed imagingmethod is based on solving the convex problem:

mins̄∈CN

‖Ψ†s̄‖1 subject to ‖y − Φs̄‖2 ≤ �, (16)

where Ψ† denotes the adjoint operator of Ψ and Φ is theNUFT operator. The alternating direction method of multi-pliers (ADMM) [46] is chosen to solve (16). The detailedimplementation of ADMM can be found in Appendix.

One important aspect in solving problem (16) is the abilityto identify the value of � that maximizes the quality of the re-construction. In the presence of Gaussian noise, a closed-formformulation of the best threshold exists based on the boundof a X 2 distribution [47]. When the noise is not Gaussian,

other methods such as least angle regression (LARS) [48],Pareto-curve-based `1- algorithms [49] and Stein unbiased riskestimator (SURE) [50] may be used.

In the case of US imaging, the noise is unknown whichmakes the use of the above methods very hard. Thus, thechoice of the best values is, in most of the studies, basedeither on cross-validation techniques or manually tuned.

The impact of the choice of � on the quality of thereconstruction will be studied in Section VI-B.

C. Compounding scheme for sparse-based beamforming

A common way to increase the contrast of PW imagingmethods is by performing compounding of PWs with varioussteering angles [10], [16]. Since speckle in the images comingfrom PWs with different angles is decorrelated, averaging theimages obtained with several PWs leads to an increase of thecontrast. In the special case of Fourier-based methods, usingcompounding also allows to populate more densely the imagespectrum and thus to reduce the interpolation error. It hasalso been described that the resolution slightly increases withcompounding [16].

Formally, let us introduce a set of T emitting angles(θti)i∈{1,..,T} and the corresponding backscattered echo sig-nals (ri)i∈{1,..,T}. The proposed method consists in consid-ering the new measurement vector which concatenates thedifferent preprocessed spectra for all the emitting anglesyC = [y1,y2, ...,yT ]

T and the new measurement operatorwhich corresponds to the concatenation of the measurementoperators for each emitting angle ΦC = [Φ1,Φ2, ...,ΦT ]

T .Then, the following problem, close to problem (16), is solved:

mins̄∈CN

‖Ψ†s̄‖1 subject to ‖yC − ΦCs̄‖2 ≤ �, (17)

This problem is solved using the same algorithm as problem(16).

V. EXPERIMENTS

A. Settings

1) Experimental settings: All the measurements have beenmade with a standard linear-probe whose settings are givenin Table I. Several PWs are emitted in order to perform PW

Parameter ValueNumber of elements (Nt) 128Center frequency (f0) 5 MHzWavelength (λ) 0.31 mmSampling frequency (fs) 31.2 MHzPitch (p) 0.193 mmKerf 0.05 mm

TABLE I: Probe characteristics.

compounding. For a desired number of PWs, the compoundingscheme used is based on steering the PW by +0.5◦ or −0.5◦starting from normal incidence. For instance, if we consider a 3PWs configuration, the angles are (−0.5◦, 0◦, 0.5◦). Constantspeed of sound is assumed (1540 m.s−1). No apodizationis used neither in transmit nor in receive. The desired RFimage is reconstructed from the backscattered echo signals

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using classical Fourier-based and delay-and-sum approachesas well as using the proposed approach. The envelope imageis extracted from the RF image through the Hilbert transform,gamma-compressed using γ = 0.3 and finally converted to8-bit gray scale to get the B-mode image.

2) Image reconstruction: For the classical approaches, azero-padding of a factor 2 in the axial direction and of a factor1.5 in the lateral direction is performed for the Fourier-basedapproaches in order to increase the image quality [37]. Thefrequency remapping is performed using a linear interpolation.

For the proposed approach, no-zero padding is performed.Four different sparsifying models are tested namely, theDirac model, the orthogonal wavelet transform (Daubechies4 wavelets), the undecimated wavelet transform (Daubechies4 wavelet) and the SA model (Daubechies 1 to Daubechies8 wavelets). Different values of the sparsifying promotingparameter � are tested ranging between 0 and ||y||2.

B. Numerical simulation

We firstly simulate the system described above using CRE-ANUIS [51], [52]. We use a cyst composed of a 8-mmdiameter anechoic occlusion centered at 4 cm depth embeddedin a medium with high density of scatterers (30 scatterers perresolution cell) whose amplitudes are distributed according toa standard normal distribution.

C. In vitro and in vivo experiments

The measurements are performed using a Verasonics ultra-sound scanner (Redmond, WA, USA) with a L12-5 50mmprobe with the same settings as the simulated probe (givenin Table I). Two types of experiments are made using thissetup. Firstly, a CIRS ultrasound phantom (Model 54GS, Com-puterized Imaging Reference Systems Inc., Norfolk, USA)is imaged. Figure 2 displays the schematic diagram of thecorresponding phantom along with the imaging plane used inthe experiment. Secondly, in vivo carotids are imaged.

Fig. 2: Schematic diagram of the CIRS phantom with theconsidered imaging plane (orange region).

VI. RESULTS

A. Choice of the sparsifying model

In order to study the effect of the sparsifying model onthe quality of reconstruction, the RF image of the simulatedphantom is reconstructed with the four sparsifying modelsdescribed in Section IV-B3, with � = 0.3||y||2 and for 1

PW insonification. The contrast [53] is calculated on thenormalized envelope image using the following formula:

CR = 20 log10|µt − µb|√

σ2t+σ2b

2

(18)

where µt and µb (σ2t , σ2b ) are the means (variances) of respec-

tively the target and the background.The contrast values, corresponding to the proposed recon-

struction coupled with the UFSB method, for the differentsparsifying models, are reported on Table II. They indicatethat the contrast is higher with the wavelet-based models thanwith the Dirac basis. This result has been expected since USimages are sparser in the wavelet-based models than in theDirac basis. Among the wavelet-based models, SA performsslightly better than the other models since it preserves a widerrange of variations of the signals [42], [45].

Dirac Orth. Wavelet Und. Wavelet SACR (dB) 7.18 8.05 8.30 8.72

TABLE II: Contrast values in dB obtained with 1 PW insonifi-cation on the simulated anechoic phantom. The reconstructionis performed with the proposed approach (UFSB method) andfour different sparsifying models.

It is known that the envelope image of diffusive specklefollows a Rayleigh distribution [54], [55]. In order to evaluatethe reconstruction of the textured area, where diffusive speckleshould be present, the goodness-of-fit against the Rayleighdistribution is tested. The envelope image is divided into non-overlapping blocks of 10×10 pixels. In each block, a one-sample Kolmogorov-Smirnov (KS) test is performed. Thistest is a widely used statistical hypothesis test that can beused to verify the equality between a sample and a referencecontinuous probability density function (pdf). In our study,the null hypothesis states that the envelope image follows aRayleigh distribution with significance level α = 0.05. For arandom variable r, the pdf of the Rayleigh distribution is given

by p (r) = rσ2 e−(r2

2σ2

)and thus requires the estimation of the

variance σ2. The parameter is estimated using the maximumlikelihood method which solution has the following closed-

form: σML =E(r2)

2 with E (r) the mean of the randomvariable r [56]. The blocks that pass the KS test are includedin the speckle region.

In order to quantify the reconstruction of the textural infor-mation, we reconstruct the simulated cyst with the proposedapproach (and UFSB) for 1 PW insonification and for thefour sparsifying models. We segment the reconstructed imagesinto two area, namely the anechoic area (pixels inside theocclusion) and the fully developed speckle area, by consideringthat the anechoic area is composed of the pixels inside thecircle centered in (0, 40) (mm) with a radius of 4 mm. Then,we calculate the number of non-overlapping blocks of 10×10pixels of the fully developed speckle area which pass the KStest for the four different sparsifying models. We divide theobtained values by the number of non-overlapping blocks of10×10 pixels present in the fully developed speckle area. The

7

S-Lu: CR = 5.46

-10 0 10

Lateral position [mm]

0

5

10

15

20

25

30

35

40

45

50

Dep

th [

mm

]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(a) � = 0.1||y||2

S-Lu: CR = 5.92

-10 0 10

Lateral position [mm]

0

5

10

15

20

25

30

35

40

45

50

Dep

th [

mm

]-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(b) � = 0.3||y||2

S-Lu: CR = 6.61

-10 0 10

Lateral position [mm]

0

5

10

15

20

25

30

35

40

45

50

Dep

th [

mm

]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(c) � = 0.5||y||2

S-Lu: CR = 8.14

-10 0 10

Lateral position [mm]

0

5

10

15

20

25

30

35

40

45

50

Dep

th [

mm

]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(d) � = 0.7||y||2

S-Lu: CR = 6.41

-10 0 10

Lateral position [mm]

0

5

10

15

20

25

30

35

40

45

50

Dep

th [

mm

]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(e) � = 0.9||y||2

Fig. 3: B-mode images of the hyperechoic CIRS phantom using the proposed method coupled with Lu method for differentvalues of � and for 1 PW insonification.

results, expressed in % and given in Table III, show thatwavelet-based models allow to recover more accurately thespeckle texture than the Dirac basis. Among the three wavelet-based models, the SA model preserves best the speckle texture.

Dirac Orth. Wavelet Und. Wavelet SA% of diffusive area 38 % 43 % 48 % 50 %

TABLE III: Percentage of diffusive speckle accurately recon-structed by the proposed approach (UFSB method) and fourdifferent sparsifying models.

In the next sections, the SA model has been chosen as thesparsifying model since it enables a slightly better reconstruc-tion than other wavelet-based models.

B. Study of the optimization parameter

In this section, the impact of the optimization parameter onthe image quality is investigated. The parameter of interestis � of equation (1) which corresponds to a higher bound ofthe distance (given by the `2-norm) between the data and thedesired solution. In this sense, the value of � quantifies therelative weight between the data fidelity constraint and thesparsity prior applied on the desired image.

In order to analyse the effect of � on the image quality,B-mode images of the hyperechoic occlusion of the CIRSphantom, present in the image plane 1, are displayed fordifferent values of � ranging from 0.1||y||2 to 0.9||y||2. Theimages, reported on Figure 3 show that when the value of� increases, the speckle density tends to decrease until thespeckle totally disappears (Fig. 3d). In this case, while the CRis maximized, it is clear that the image is over-regularized.

To illustrate what we mentioned before, Figure 4 displaysthe evolution of both the contrast and the percentage ofrecovered diffuse speckle area, calculated with the sameprocess as in Section VI-A, for an increasing value of thesparsity promoting parameter. It can be observed that, whilethe contrast keeps increasing, the speckle density tends todecrease when sparsity in the SA model is promoted in thereconstruction.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Values of the sparsity promoting parameter

0

10

20

30

40

50

60

70

80

90

100

% o

f fu

lly d

eve

lop

ed

sp

eckle

Ratio fully developed speckle

CR

5

6

7

8

9

10

Co

ntr

ast

Ra

tio

(d

B)

Fig. 4: Evolution of the contrast ratio (red) and the per-centage of recovered speckle density for increasing valuesof the sparsity promoting parameter. The metrics have beencalculated on the images obtained with 1 PW insonificationand reconstructed using the sparse-based approach with Lumethod.

The study shows that part of the diffusive speckle is notsparse enough in the proposed model and then considered asnoise. It is clear that the choice of the right parameter highlydepends on both the image content and the user needs. Ifspeckle is considered as noise, then it can be removed byforcing sparsity in a wavelet-based model. If both speckleand texture have to be preserved, a low value of � (between0.1||y||2 and 0.3||y||2 for Figure 3) seems to be a goodtrade-off between an increase of the contrast and a reasonablespeckle density.

C. Comparison against classical approaches

1) Contrast: Figure 5 displays the CR values for theproposed method and for the state of the art methods for eachcompounding experiment, with � = 0.3||y||2. It shows that theproposed method leads to an increase of the contrast of morethan 2.5 dB for 1 PW. This gap decreases with compoundingsince the CR increases far faster for the classical methods thanfor the proposed method. This can be explained by the factthat compounding decreases the noise level by decorrelating

8

1 3 5 7 9 11 13 15

Number of plane waves

5

6

7

8

9C

R [

dB

]

UFSB

Garcia

Lu

Montaldo

S-UFSB

S-Garcia

S-Lu

Fig. 5: Contrast ratio as a function of the number of PWsfor the different reconstruction methods. The dashed linesrepresent the classical methods and the solid lines representthe sparse-based method.

the random speckle [10]. The proposed method also removesthe measurement noise by enforcing sparsity in the SA model.Thus, the two methods inducing the same consequence, theimpact of the compounding in terms of noise removal is lessimportant. However, it is interesting to note that the proposedmethod, with 1 PW, leads to higher contrasts than classicalmethods with 15PWs, thus achieving a noticeable reductionof the measurements needed to reach a given image quality.

Figure 6 confirms what we deduced from the contrastmeasurement. The proposed method removes the noise insidethe inclusion which appears with the classical methods. It canalso be observed that the speckle density slightly decreasesin the far field for the proposed method. This aspect will bediscussed in Section VI-C3.

Since the contrast is measured in the anechoic area, it isdirectly linked to the amount of noise in the image and sincethe simulation is noiseless, the only source of noise in theexperiment is induced by the approximation of the measure-ment model. The proposed approach leads to an increase ofthe contrast thus to a decrease of the noise created by themeasurement model, which means that the proposed measure-ment process is more accurate than the classical filtered backprojection.

2) Resolution: The lateral resolution is calculated on the B-mode image as the width at -6 dB of the point spread function(PSF) of the two points at 2cm and 4cm of the CIRS phantomwhich corresponds to the full width at half maximum [53].Figure 7 displays the evolution of the lateral resolution withthe number of PWs for the different reconstruction methods.

From Figure 7, it can be observed that the proposed methodleads to a slight increase of the image resolution. This maybe justified by the fact that the proposed method significantlydecreases the magnitude of the side lobes of the PSF.

From Figure 7a, it can be noticed that the proposed approachgives results similar to the classical approach except for Lumethod in which the resolution is better with the proposedapproach.

From Figure 7b, it can be noticed that the proposed methodleads to a slight increase of the resolution for UFSB. For

Garcia and Lu methods, the resolution is rather similar. ForGarcia method, it can also be observed that the resolutiondecreases when the number of PWs increases. While beingcounter-intuitive, this result is in accordance with what Garciaet al. observed [16].

On Figure 8, the effect of the proposed method on thePSF is investigated. It can be observed that the proposedmethod drastically decreases the magnitude of the side lobes.This observation is in accordance with the results observed inSection VI-C.

3) Speckle density: On Figure 6, it can be noticed that thespeckle density seems to be lower for the proposed approachthan for the classical methods. In order to quantify the impactof the proposed approach on the texture, the same procedureas for the texture experiment in Section VI-A is followed forboth the proposed approach and the classical methods. Thepercentages of the total diffusive speckle area recovered withthe classical methods are 66% for Lu method, 59% for UFSBand 62% for Garcia method. These results are around 10%higher than the ones obtained with the proposed approach(Table III). The difference is justified by the fact that specklehas a very complex behaviour, which is different at each depth.This complex structure is very hard to preserve and fullydeveloped speckle is not sufficiently sparse even in complexwavelet-based models. This aspect will be discussed in SectionVII-B.

D. In vivo experiments

The proposed approach is finally evaluated on in vivo carotidimages. Since the ground truth is not known, the use of themetrics defined in Section VI-A is not anymore possible.

In the experiments, the images are reconstructed with only1 PW insonification. The sparsity promoting parameters havebeen manually tuned based on visual evaluation of the re-constructed image. From Figure 9, it can be seen that partof the noise in the carotid artery and between 5 mm and15 mm have been removed. Moreover, the proposed methodreduces the side lobes as it can be seen particularly close tothe upper carotid wall. However, as expected from the studyof Section VII-B, the speckle density in the far field (fartherthan 20mm) is lower with the proposed approach than withclassical methods.

As a reference, Figure 10 displays the B-mode image of theDAS reconstruction obtained with 15 PWs. It can be seen thatthe proposed reconstruction leads to a visual quality closer tothe one of Figure 10 than the classical methods.

VII. DISCUSSION

A. The use of sparse regularization to solve the problem

The use of sparse regularization to solve the ill-posed prob-lem mainly comes from CS framework and the assumption thatUS images are compressible in well chosen models. One canquestion the use of sparsity prior as an appropriate regularizerfor the problem. Another alternative may be to use `2-normand perform an inversion similar to Wiener filtering [57].The main advantage of this approach resides in its simpleimplementation since a closed form formula exists for the

9

S-UFSB: CR = 8.72

-10 0 10

Lateral position [mm]

30

35

40

45

50

Dep

th [

mm

]

(a)

S-Garcia: CR = 8.57

-10 0 10

Lateral position [mm]

30

35

40

45

50

De

pth

[m

m]

(b)

S-Lu: CR = 8.47

-10 0 10

Lateral position [mm]

30

35

40

45

50

De

pth

[m

m]

(c)

UFSB: CR = 6.04

-10 0 10

Lateral position [mm]

30

35

40

45

50

De

pth

[m

m]

(d)

Garcia: CR = 5.73

-10 0 10

Lateral position [mm]

30

35

40

45

50

De

pth

[m

m]

(e)

Lu: CR = 6.04

-10 0 10

Lateral position [mm]

30

35

40

45

50

Dep

th [

mm

]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(f)

Fig. 6: B-mode images of the simulated occlusion using the proposed method with (a) UFSB, (b) Garcia and (c) Lu and usingthe classical method with (d) UFSB, (e) Garcia and (f) Lu for 1 PW insonification.

1 3 5 7 9 11 13 15

Number of plane waves

0.4

0.6

0.8

1

1.2

1.4

Late

ral re

so

luti

on

[m

m]

(a)

1 3 5 7 9 11 13 15

Number of plane waves

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

La

tera

l re

so

luti

on

[m

m]

UFSB

Garcia

Lu

Montaldo

S-UFSB

S-Garcia

S-Lu

(b)

Fig. 7: Lateral resolution measured on the points of the CIRS phantom located at a depth of (a) 2 cm and (b) 4 cm for differentcompounding experiments.

7 9 11

Lateral dimension (mm)

-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de

(d

B)

UFSB

S-UFSB

(a)

7 9 11

Lateral dimension (mm)

-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de

(d

B)

Garcia

S-Garcia

(b)

7 9 11

Lateral dimension (mm)

-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de

(d

B)

Lu

S-Lu

(c)

Fig. 8: Lateral point spread function measured on the point of the CIRS phantom located at a depth of 2 cm, for 1 PWinsonification, and reconstructed with (a) UFSB, (b) Garcia and (c) Lu. The red dashed line represents the proposed methodand the blue line corresponds to the classical method.

`2-regularization problem. Nonetheless, the perfect recoverycondition that holds for `1-regularization does not remain validfor `2-regularization. Additionally, in practice, the quality ofthe reconstruction (in terms of contrast and resolution) is lowerwith `2-regularization than with `1-regularization methodssince Wiener filtering tends to smooth the information in theultrasound images [58], [59] .

The `1-norm is used in the optimization problem as sparsitypromoting norm. The choice of such norm against `p-normwith p < 1 is justified firstly by the convexity of the optimiza-tion problem, leading to a unique solution and secondly by the

availability of an extensive number of methods to solve suchproblem. One can suggest the use of reweighted `1-algorithms[45], [60] as an alternative to the proposed approach. Whilebeing a better approximate of the `0-norm, it requires manyiterations of the optimization algorithm. Our main motivationof using `1- algorithms instead of reweighting `1-algorithmsis convergence time.

Several studies have used `p-norm with p ∈ ]1, 2] insteadof `1-norm due to the statistical behaviour of scatterers map,closer to Generalized Gaussian model than to Laplacian model[32], [57]. In the proposed approach, the choice of the `1-norm

10

S-UFSB

-10 -5 0 5 10

Lateral position [mm]

5

10

15

20

25

Dep

th [

mm

]

(a)

S-Garcia

-10 -5 0 5 10

Lateral position [mm]

5

10

15

20

25

Dep

th [

mm

]

(b)

S-Lu

-10 -5 0 5 10

Lateral position [mm]

5

10

15

20

25

De

pth

[m

m]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(c)UFSB

-10 -5 0 5 10

Lateral position [mm]

5

10

15

20

25

Dep

th [

mm

]

(d)

Garcia

-10 -5 0 5 10

Lateral position [mm]

5

10

15

20

25

Dep

th [

mm

]

(e)

Lu

-10 -5 0 5 10

Lateral position [mm]

5

10

15

20

25

Dep

th [

mm

]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

(f)

Fig. 9: B-mode images of the carotid phantom using the proposed method with (a) UFSB, (b) Garcia and (c) Lu and usingthe classical method with (d) UFSB, (e) Garcia and (f) Lu for 1 PW insonification.

Montaldo

-10 -5 0 5 10

Lateral position [mm]

5

10

15

20

25

De

pth

[m

m]

-40 dB

-30 dB

-20 dB

-10 dB

0 dB

Fig. 10: B-mode image of the DAS reconstruction with 15PWs.

in a given model is motivated by geometrical considerationsmore than statistical prior. In the above mentioned approaches,the US image is deconvolved and statistical priors are assumedon the deconvolved image, usually called tissue reflectivityfunction (TRF) or scatterers map. In the proposed approach,the unknown image is the RF image which does not neces-sarily exhibit the same statistical behaviour. This differenceis described by Chen et al. [32] who present a compresseddeconvolution framework in which a sparsity prior in thewavelet domain is used for the RF image and GeneralizedGaussian Distribution prior is used for the TRF.

With the current implementation of the inverse problem, theresolution is not improved. However, the proposed approach is

compatible with the state-of-the-art deconvolution frameworkssuch as the compressive deconvolution framework [32]. Bycombining the two approaches, it would be possible to improvethe resolution.

B. Speckle density

While it has been demonstrated that a wavelet-based modelenables better capturing textural information of images thana Dirac basis, part of the speckle remains being consideredas measurement noise, showing that fully developed speckleis not entirely sparse in wavelet-based models. This is alimitation of the proposed approach and we can think abouttwo ways to overcome these drawback. The first one is toperform region-based optimization. The idea would be tosegment US images in different regions either in a very simpleway (by dividing the image into blocks of fixed size) orwith more elaborated segmentation methods based on struc-tural information inside regions [61]. Then, the optimizationproblems are solved on each region independently allowingto adapt the reconstruction to the content of each region.The main drawback of such optimization is the computationalcomplexity as well as the need for a stitching method toreconstruct the full image from the different regions. The otheralternative resides in performing dictionary learning [33]. Withsuch approach, the analysis model is learnt on a training setand a sparser representation of the speckle may be found. Thiswould very probably leads to better results. Nonetheless, thisapproach requires an extensive number of ultrasound images

11

in the training set in order to cover all the possible casesof diffusive speckle and the related analysis model is verycomplex.

C. Computational complexity

Solving `1-regularized problems to reconstruct the desiredimage usually involves non-linear iterative algorithms. Thus,computational complexity is a problem to take into accountwhen using these methods. Indeed, considering the schemeof the ADMM algorithm in appendix, it can be seen thatthe algorithm is composed of a gradient descent and a soft-thresholding in the analysis domain. This implies two matrixproducts with Φ and Φ† for the gradient descent and twomatrix products with Ψ† and Ψ, respectively, for the soft-thresholding operation. Since the matrix Ψ is made of aconcatenation of wavelet bases, its size is several times thesize of the image, depending on the number of wavelet basesconsidered. Doing the matrix product is then costly. A way toaddress such problem is by considering parallel implemen-tation of wavelets on GPU and to exploit fast algorithmsto implement the wavelet decomposition with O(N logN)complexity.

Regarding the matrix Φ, the NUFFT algorithm proposedby Fessler and Sutton [62] consists in a zero padded 2DFFT followed by an interpolation step to compute the valuesoutside the regular grid. The operator can be modelled asΦ = GFZ, where Z denotes the zero padding operator, Fdenotes the 2D FFT operator (in the upsampled size) and G isa sparse matrix implementing the convolutional interpolationoperator. Considering that the desired image is composed ofN pixels, that the number of resampled frequency locations isK = 2N (typical value), that M is the number of non-uniformfrequencies and that the interpolation neighbourhood is J , ithas been demonstrated that the computational complexity ofthe NUFFT is O(K logN +JM) [37]. Since it depends bothon the number of non-uniform frequency samples and on thenumber of points in the final image, it is far more complexthan the FFT and becomes extremely slow when used in acompounding scheme where the total number of frequencies,M , is large.

One alternative to speed up the reconstruction algorithm isto grid the measured data onto a regular grid, i.e. y′ = GTy,and approximate the NUFFT by an FFT followed by mask onthe sensed discrete frequencies, i.e. Φ = MF, where F denotesthe 2D FFT of size N and M is a Q × N diagonal binarymatrix that selects the sensed discrete frequencies. Indeed, thelater model approximates the holographic matrix GTG, thatgrids the continuous frequency samples back to the uniformgrid, by the diagonal binary matrix M. This simplified modelallows us to drastically decrease the complexity of the operatorΦ while preserving the reconstruction quality almost intact.

D. Fourier-based beamforming methods

To the best of our knowledge, Fourier-based beamformingmethods have been investigated by Chernyakova et al. [35]. Intheir work, they introduced the beamforming in the frequencydomain as an alternative to the classical DAS beamforming.

The objective is to exploit the band-limited properties of USsignals in order to drastically reduce the data rate comparedto classical approach.

In a second step, they suggest to use CS in order tofurther decrease the data rate. In this case, the Fourier seriescoefficients of the beamformed signal are expressed as asum of phase-shifted pulse shapes, with a given amplitude,through a finite rate of innovation model. CS framework isused to retrieve the pulse amplitudes from the Fourier seriescoefficients.

Thus, the work proposed by Chernyakova et al. is, byessence, different from the proposed approach. First, the objec-tive is different since the proposed work focuses on improvingthe results of the plane wave reconstruction while the workof Chernyakova et al. emphasizes the data rate reduction.Moreover, while the work of Chernyakova et al. exploits CSin order to retrieve pulses amplitudes later used to reconstructthe RF image, the proposed work uses CS as a way to recoverthe RF image coefficients directly.

VIII. CONCLUSIONIn this paper, a novel framework for Fourier-based recon-

struction of signals obtained with several PW insonificationshas been proposed. The framework relies on the ability topose the Fourier reconstruction problem as an ill-posed inverseproblem and and on the sparsity of the US images in ananalysis domain. The reconstruction is achieved by solvingan `1-minimization problem. Different sparsifying transformshave been studied and the SA model exhibits better recon-struction results than other wavelet-based models. Simulationsand experiments enlighten a better image quality (contrast) forthe proposed approach than for the state-of-the-art methodswith a slight decrease of the speckle density. This decreasecomes from the fact that fully developed speckle is not entirelysparse in the proposed analysis models. However, the proposedframework opens the door to a variety of promising applica-tions such as compressed beamforming and deconvolution.

ACKNOWLEDGEMENTSThis work was supported in part by the UltrasoundToGo

RTD project (no. 20NA21 145911), evaluated by the SwissNSF and funded by Nano-Tera.ch with Swiss Confederationfinancing. This work was also performed within the frameworkof the LABEX PRIMES (ANR- 11-LABX-0063) of Univer-site de Lyon, within the program ”Investissements d’Avenir”(ANR-11-IDEX-0007) operated by the French National Re-search Agency (ANR). The RF Verasonics generator wascofounded by the FEDER program, Saint-Etienne Metropole(SME) and Conseil General de la Loire (CG42) within theframework of the SonoCardio- Protection Project leaded byPr Pierre Croisille.

APPENDIX: ADMM ALGORITHMThe pseudo-code of the ADMM algorithm used in the

proposed method is given below. The general problem wesolve is the following one:

minx∈CN ,z∈CM

f (x) + h (z) subject to Φx+ z = y, (19)

12

given the assumption that f : CN → R and h : CM → R arelower semicontinuous convex functions.

The key mathematical tool used in ADMM is the proximityoperator of a convex function defined as:

proxf (x) = arg miny∈CM

f (y) +1

2||y − x||22. (20)

In the proposed `1-minimization problem f(x) = ‖Ψ†x‖1and h(z) = iB(z), where iB is the indicator function of theconvex set B defined as B = {z ∈ CM |‖z‖2 ≤ �}.

The general structure of the algorithm is detailed in Algo-rithm 1. The parameters µ > 0 and β > 0 are step sizeschosen such that µL + β < 2, where L is the spectral normof the matrix Φ, and γ > 0 is a thresholding constant thatcontrols the convergence speed.

Algorithm 1 ADMM algorithm

Require: t = 0, choose x0, z0, λ0, γ, µ and β.repeatz(t+1) = proxγh(y − Φx(t) − λ(t))s(t+1) = x(t) − µΦH

(λ(t) + Φx(t) − y + z(t+1)

)x(t+1) = proxµγf

(s(t+1)

)λ(t+1) = λ(t) + β

(Φx(t+1) − y + z(t+1)

)until A stopping criterion is met

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